PUB - Publikationen an der Universität Bielefeld

We consider the product of n complex non-Hermitian, independent randommatrices, each of size NxN with independent identically distributed Gaussianentries (Ginibre matrices). The joint probability distribution of the complexeigenvalues of the product matrix is found to be given by a determinantal pointprocess as in the case of a single Ginibre matrix, but with a more complicatedweight given by a Meijer G-function depending on n. Using the method oforthogonal polynomials we compute all eigenvalue density correlation functionsexactly for finite N and fixed n. They are given by the determinant of thecorresponding kernel which we construct explicitly. In the large-N limit atfixed n we first determine the microscopic correlation functions in the bulkand at the edge of the spectrum. After unfolding they are identical to that ofthe Ginibre ensemble with n=1 and thus universal. In contrast the microscopiccorrelations we find at the origin differ for each n and generalise the knownBessel-law in the complex plane for n=2 to a new hypergeometric kernel 0_F_n-1.